The non-commutative analogue of the difference and differential systems for \(a_n\)  and \(b_n\)  are given here in Theorems REF  and REF . The derivation, which is based upon the Riemann–Hilbert method established in [1]}, is not a straightforward generalisation of the one for the scalar case: namely one has to push a little bit further the analysis of the behavior of the solution of the Riemann–Hilbert problem at the two singular points 0 and \(\infty \) . Also, while the systems of \(a_n\)  and \(b_n\)  were of first-order, here our two systems are of second-order, which is a phenomenon that, in some way, we already observed in the previous paper [2]} (and also, to some extent, in [3]} for the case of Painlevé XXXIV).
